Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.6 Half-Angle Identities - 5.6 Exercises - Page 244: 64


$\theta = 54^{\circ}$

Work Step by Step

Consider the right triangle with an angle of $\frac{\theta}{2}$. Let $h$ be the hypotenuse, and let $a$ be the adjacent side. Note that $h = a+b$. We can find $a$: $h^2 = a^2+50^2$ $(a+b)^2 = a^2+50^2$ $a^2+2ab+b^2 = a^2+50^2$ $2ab+b^2 = 50^2$ $a = \frac{50^2-b^2}{2b}$ $a = \frac{50^2-(12)^2}{(2)(12)}$ $a = 98.17$ We can find $\frac{\theta}{2}$: $tan~\frac{\theta}{2} = \frac{50}{98.17}$ $\frac{\theta}{2} = arctan(\frac{50}{98.17})$ $\frac{\theta}{2} = 27^{\circ}$ Therefore, $\theta = 54^{\circ}$
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